Solve $LaTeX: \displaystyle \log_{12}(x + 1019)+\log_{12}(x + 238) = 5$ .

Using logarithmic properties and expanding the argument gives $LaTeX: \displaystyle \log_{12}(x^{2} + 1257 x + 242522)=5$ . Making both sides an exponent on the base gives $LaTeX: \displaystyle x^{2} + 1257 x + 242522=12^{5}$ . Expanding and setting equal to zero gives $LaTeX: \displaystyle x^{2} + 1257 x - 6310=0$ . Factoring gives $LaTeX: \displaystyle \left(x - 5\right) \left(x + 1262\right)=0$ . Solving gives the two possible solutions $LaTeX: \displaystyle x = -1262$ and $LaTeX: \displaystyle x = 5$ . The domain of the original is $LaTeX: \displaystyle \left(-1019, \infty\right) \bigcap \left(-238, \infty\right)=\left(-238, \infty\right)$ . Checking if each possible solution is in the domain gives: $LaTeX: \displaystyle x = -1262$ is not a solution. $LaTeX: \displaystyle x=5$ is a solution.